1. Field of the Invention
The present invention relates to a wireless telecommunication system, and more particularly to a wireless telecommunication system using multiple antennas at a transmitter and multiple antennas at a receiver, also called multiple-input multiple-output (MIMO) system.
2. Discussion of the Background
MIMO telecommunication systems are well known in the prior art for their capability of achieving dramatically improved capacity as compared to single antenna to single antenna or multiple antennas to single antenna systems.
The principle of a MIMO telecommunication system has been illustrated in FIG. 1. The data stream Din to be transmitted is encoded at the transmitter side by a vector encoder 110 and mapped into a plurality of symbol substreams, each substream being dedicated to a given antenna. The different substreams are RF modulated and amplified by the Tx units 1201, 1202 . . . , 120M before being transmitted by the antennas 1301, 1302, . . . , 130M. At the receiver side, a plurality of antennas 1401, 1402, . . . , 140N receive the transmitted signals and the received signals are RF demodulated into symbols by the Rx units 1501, 1502 . . . , 150N. The symbols thus obtained are processed by a detector 160 to produce a stream of received data Dout.
Various schemes have been proposed in the prior art for the vector encoder 110 and the detector 160. The basic idea underlying these schemes is to exploit both the space diversity (due to the different fading coefficients affecting the propagation between the transmit and receive antennas) and time diversity. For this reason, the unit 110 is also called a space-time encoder. For example, a space-time diversity technique has been proposed in an article of Tarokh et al. entitled “Space-time codes for high data rates wireless communications: performance criterion and code construction” published in IEEE Trans. Inform. Theory, vol. 44, pp 744-765, March 1998. This technique, named STTC (for Space-Time Trellis Coding) creates inter-relations between transmitted signals in the space domain (different transmit antennas) and the time domain (consecutive time symbols) according to a trellis structure. The transition in the trellis are determined by the input symbols. The detector 160 is based on a Maximum Likelihood Sequence Estimation (MLSE), computing the lowest accumulated metric to determine the most likely transmitted sequence. Another space-time diversity technique has been proposed by S. M. Alamouti et al. in an article entitled “A simple transmit diversity technique for wireless communications” published in IEEE J. Select. Areas. Comm., vol. 16, pp 1451-1458, October 1998. According to this technique, called STBC (for Space-Time Block Coding), a block of input symbols is mapped into a L×M coded matrix where L is a number of consecutive time slots and the L vectors constituting said matrix are transmitted by the M transmit antennas.
More recently, it has been shown by O. Damen et al. in an article entitled “Lattice code decoder for space-time codes” published in IEEE Communication Letters, vol. 4, no5, pp 161-163, May 2000, that the signal received a MIMO telecommunication system can be represented as a point of a lattice corrupted by noise and that, accordingly, a sphere decoder can be used at the receiver side for obtaining a ML estimation of the transmitted symbol vector. More precisely, the sphere decoding method has been proposed for both an uncoded MIMO system, i.e. a system in which the substreams of symbols associated with the different transmit antennas are coded independently and for a MIMO system using so-called algebraic space-time codes. An example of such codes can be found in the PhD. thesis of O. Damen entitled “Joint coding/decoding in a multiple access system—Application to mobile communications” ENST, Paris.
Let us consider a MIMO telecommunication system as represented in FIG. 1 and let us denote x(p)=(x1(p), x2(p), . . . , xM(p)) and y(p)=(y1(p), y2(p), . . . , yN(p)) respectively the vector of symbols transmitted by the antennas and the vector of signals received by the antennas at a given time p. We suppose first that the MIMO system the substreams of symbols have been independently coded. We may write:
                                          y            n                    ⁡                      (            p            )                          =                                            ∑                              m                =                1                            M                        ⁢                                                            h                  mn                                ⁡                                  (                  p                  )                                            ⁢                                                x                  m                                ⁡                                  (                  p                  )                                                              +                                    η              n                        ⁡                          (              p              )                                                          (        1        )            where hmn(p) is the fading coefficient for the propagation path between transmit antenna m and receive antenna n at time p and ηn(p) is the noise sample affecting the received signal yn(p). In general, the transmission is based on frames of length L and the coefficients are supposed constant over a transmission frame but may vary from frame to frame, e.g. the M×N transmission channels from the M transmit antennas to the N receive antennas are supposed to be quasi-static Rayleigh channels. The fading coefficients are supposed to be known at the receiver, for example thanks to an estimation of pilot symbols transmitted by the different transmit antennas. The noise samples are supposed to be samples of independent complex Gaussian variables with zero mean (AWGN noise) and variance σ2.
It should be noted that the transmitted symbols does not necessarily belong to the same modulation constellation.
The expression (1) can be rewritten by adopting a matrix formulation and omitting the time index p:y=xH+η  (2)where H is a M×N matrix defined by H=(hmn), m=1, . . . , M; n=1, . . . , N.
The complex components of the vectors x, y and η in equation (2) can respectively be expressed as xm=xmR+jxmI, ym=ymR+jymI, ηm=ηmR+jηmI. Similarly the complex coefficients hmn of the matrix H can be expressed as
      h    mn    =            h      mn      R        +          j      ⁢                          ⁢              h        mn        I            where
      h    mn    R    ⁢          ⁢  and  ⁢          ⁢      h    mn    I  are real.
Denoting x′=(x1R,x1I, . . . , xMR, xMI), y′=(y1R, y1I, . . . , yNR, yNI), η′=(η1R, η1I, . . . , ηNR, ηNI)
            and      ⁢                          ⁢              H        ′              =          [                                                  h              11              R                                                          h              11              I                                            ⋯                                              h                              1                ⁢                N                            R                                                          h                              1                ⁢                N                            I                                                                          -                              h                11                I                                                                        h              11              R                                            ⋯                                              -                              h                                  1                  ⁢                  N                                I                                                                        h                              1                ⁢                N                            R                                                            ⋮                                ⋮                                                                                          ⋮                                ⋮                                                              h              M1              R                                                          h              M1              I                                            ⋯                                              h              MN              R                                                          R              MN              I                                                                          -                              h                M1                I                                                                        h              M1              R                                            ⋯                                              -                              R                MN                I                                                                        R              MN              R                                          ]        ,we can rewrite the expression (2) as:y′=x′H′+η′  (3)where y′ and η′ are 1×2N real vectors, x′ is a 1×2M real vector and H′ is a 2M×2N real matrix. Without loss of generality, it will be supposed in the sequel that the real and imaginary components xmR and xmI of each transmitted symbol xm are PAM modulated, i.e.:xmR∈{−M2m−1+1,−M2m−1+3, . . . , M2m−1−3,M2m−1−1} and  (4)xmI∈{−M2m+1, −M2m+3, . . . , M2m−3, M2m−1}  (5)where M2m−1 and M2m are the modulation orders for xmR and xmI respectively. For example if xm is a 16-QAM modulated symbol M2m−1=M2m=4.
The following results can be extended to the case where the transmitted symbols are PSK modulated, as shown in the article of Hochwald et al. entitled “Achieving near-capacity on a multiple antenna channel” available on the website mars.bell-labs.com.
If the following affine transformation is effected:
                    x        ~            m      R        =                            1          2                ⁢                  (                                    x              m              R                        +                          M                                                2                  ⁢                  m                                -                1                                      -            1                    )                ⁢                                  ⁢        and        ⁢                                  ⁢                                                                                        ⁢              x                        ~                    m          I                    =                        1          2                ⁢                  (                                    x              m              I                        +                          M                              2                ⁢                m                                      -            1                    )                      ⁢        or again vectorially:
                              x          ~                =                              1            2                    ⁢                      (                                          x                ′                            +              μ                        )                                              (        6        )            where {tilde over (x)}=({tilde over (x)}1R,{tilde over (x)}1I, . . . , {tilde over (x)}MR, {tilde over (x)}MI) and μ=(M1−1, M2−1, . . . , M2M−1). The components of {tilde over (x)} are elements of Z and consequently {tilde over (x)} is a vector of Z2M.
In general terms, there exists an affine transform transforming the components xmR and xmI into elements of Z and the vector {tilde over (x)} can consequently be represented as a vector of Z2M.
In a similar manner, the corresponding transform is effected on y′ as defined in (3), that is to say:
                              y          ~                =                              1            2                    ⁢                      (                                          y                ′                            +                              μ                ⁢                                                                  ⁢                                  H                  ′                                                      )                                              (        7        )            
The vector {tilde over (y)} in equation (7) can therefore be expressed as:{tilde over (y)}={tilde over (x)}H′+η/2  (8)
We suppose that M≦N (i.e. the number of receive antennas is larger than the number of transmit antennas) and rank(H′)=2M, which is the case in practice if the fading coefficients are decorrelated. The following results can however be extended to the case where M>N as shown in the article of O. Damen et al. entitled “A generalized sphere decoder for asymmnetrical space-time communication architecture” published in Elec. Letters, vol. 36, pp. 166-168, January 2000. Indeed, in such instance, an exhaustive search can be effected for M−N symbols among the M transmitted symbols, the remaining N symbols being searched by the sphere decoding method as described further below.
The vector {tilde over (y)} can be regarded as a point of a lattice Λ of dimension 2M in R2Nand of generator matrix H′, corrupted by a noise η′/2. Indeed, a lattice Λ of dimension κ in a vectorial space of dimension K is defined by any set of vectors v of Rκ satisfying:v=b1v1+b2v2+ . . . +bκvκ  (9)where {v1, v2, . . . , vκ} are linearly independent vectors of RK and b=(b1, . . . , bκ)∈Zκ.
The vectors v1, v2, . . . , vκ form the rows of the so-called generator matrix G of the lattice., and are therefore called the generating vectors of said the lattice. It is therefore possible to write:v=bG  (10)
Such a lattice has been represented in FIG. 2A in the simple case κ=2. In general, the dimension of the lattice Λ generated by G=H′ is κ=2M in a space of dimension K=2N.
The set of transmitted symbols can be represented by an alphabet of finite size Π⊂Z2M hereinafter referred to as a product constellation. This product constellation is determined by the modulation constellations used for modulating the M symbol substreams and the cardinal of the alphabet Π is equal to the product of the cardinals of the different modulation alphabets. The product constellation Π corresponds to a subset of the lattice Λ.
An exhaustive maximum likelihood decoding would require a search for the closest neighbour throughout the product constellation Π, i.e. would require to search for z∈Π so that the distance ∥z−{tilde over (y)}∥ is minimum (in the representation of FIG. 2A).
The sphere decoding method calculates the distances to the points which are situated within an area of the lattice situated around the received point, preferably inside a sphere S of given radius √{square root over (C)} centered on the received point as depicted in FIG. 2A. Only the points in the lattice situated at a quadratic distance less than C from the received point are therefore considered for the minimization of the metric.
In practice, the decoder effects the following minimization:
                                          min                          z              ⁢                                                          ⁢              ε              ⁢                                                          ⁢              Λ                                ⁢                                                z              -                              y                ~                                                                =                              min                                          w                ⁢                                                                  ⁢                ε                ⁢                                                                  ⁢                                  y                  ~                                            -              Λ                                ⁢                                  w                                                          (        11        )            
To do this, the smallest vector w in the translated set {tilde over (y)}−Λ is sought. If we denote ρ={tilde over (y)}H′ and ξ=wH′ where H′ is the pseudo-inverse (also called the Moore-Penrose inverse) of the matrix H′, the vectors {tilde over (y)} and w can be expressed as:{tilde over (y)}=ρH′ with ρ=(ρ1, . . . , ρ2M) w=ξH′ with ξ=(ξ1, . . . , ξ2M)  (12)
It is important to note that ρ and ξ are both real vectors. According to expression (8) the vector ρ can be regarded as the ZF estimation of the vector {tilde over (x)}. As w={tilde over (y)}−z, where z=bH′, belongs to the lattice Λ, we have ξi=ρi−bi for i=1, . . . , 2M with
  w  =            ∑              i        =        1                    2        ⁢        M              ⁢                  ⁢                  ξ        i            ⁢                        v          i                .            The vector w is a point in the lattice whose coordinates ξi are expressed in the translated reference frame centered on the received point {tilde over (y)}. The vector w belongs to a sphere of quadratic radius C centered at 0 if and only if:∥w∥2=Q(ξ)=ξH′H′TξT≦C  (13)
In the new system of coordinates defined by ρ, b and ξ the sphere of quadratic radius C centered at {tilde over (y)} is therefore transformed into an ellipsoid E and the lattice Λ is represented as elements of Z2M. This representation is illustrated in FIG. 2B. It should be understood that it is equivalent to the one illustrated in FIG. 2A.
The Cholesky factorization of the Gram matrix Γ=H′H′T gives Γ=ΔΔT, where Δ is an inferior triangular matrix of elements δij. The expression (13) can be rewritten:
                              Q          ⁡                      (            ξ            )                          =                              ξ            ⁢                                                  ⁢            Δ            ⁢                                                  ⁢                          Δ              T                        ⁢                          ξ              T                                =                                                                                                          Δ                    T                                    ⁢                                      ξ                    T                                                                              2                        =                                                            ∑                                      i                    =                    1                                                        2                    ⁢                    M                                                  ⁢                                                      (                                                                                            δ                          ii                                                ⁢                                                  ξ                          i                                                                    +                                                                        ∑                                                      j                            =                                                          i                              +                              1                                                                                                            2                            ⁢                            M                                                                          ⁢                                                                              δ                                                          j                              ⁢                                                                                                                          ⁢                              i                                                                                ⁢                                                                                                          ⁢                                                      ξ                            i                                                                                                                )                                    2                                            ≤              C                                                          (        14        )            By putting:
                                                        q              ii                        =                                                            δ                  ii                  2                                ⁢                                                                  ⁢                for                ⁢                                                                  ⁢                i                            =              1                                ,          …          ⁢                                          ,                      2            ⁢            M                          ⁢                                  ⁢                                            q              ij                        =                                                                                δ                    ij                                                        δ                    jj                                                  ⁢                                                                  ⁢                for                ⁢                                                                  ⁢                j                            =              1                                ,          …          ⁢                                          ,                                    2              ⁢              M                        ;                                                  ⁢                          i              =                              j                +                1                                              ,          …          ⁢                                          ,                      2            ⁢            M                                              (        15        )            there is obtained:
                              Q          ⁡                      (            ξ            )                          =                              ∑                          i              =              1                                      2              ⁢              M                                ⁢                                          ⁢                                                    q                ii                            ⁡                              (                                                      ξ                    i                                    +                                                            ∑                                              j                        =                                                  i                          +                          1                                                                                            2                        ⁢                        M                                                              ⁢                                                                  q                        ji                                            ⁢                                              ξ                        j                                                                                            )                                      2                                              (        16        )            
Being concerned first of all with the range of possible variations of ξ2M, and then adding the components one by one by decreasing index order, the following 2M inequalities are obtained, which define all the points within the ellipsoid:
                                                        q                                                2                  ⁢                  M                                ,                                  2                  ⁢                  M                                                      ⁢                          ξ                              2                ⁢                M                            2                                ≤          C                ⁢                                  ⁢                                                                              q                                                                                    2                        ⁢                        M                                            -                      1                                        ,                                                                  2                        ⁢                        M                                            -                      1                                                                      ⁡                                  (                                                            ξ                                                                        2                          ⁢                          M                                                -                        1                                                              +                                                                  q                                                                              2                            ⁢                            M                                                    ,                                                                                    2                              ⁢                              M                                                        -                            1                                                                                              ⁢                                              ξ                                                  2                          ⁢                          M                                                                                                      )                                            2                        +                                          q                                                      2                    ⁢                    M                                    ,                                      2                    ⁢                    M                                                              ⁢                              ξ                                  2                  ⁢                  M                                2                                              ≤          C                ⁢                                  ⁢                              ∀                          l              ∈                              {                                  1                  ,                  …                  ⁢                                                                          ,                                      2                    ⁢                    M                                                  }                                              ,                                                    ∑                                  i                  =                  l                                                  2                  ⁢                  M                                            ⁢                                                                    q                    ii                                    ⁡                                      (                                                                  ξ                        i                                            +                                                                        ∑                                                      j                            =                                                          i                              +                              1                                                                                                            2                            ⁢                            M                                                                          ⁢                                                                              q                            ji                                                    ⁢                                                      ξ                            j                                                                                                                )                                                  2                                      ≤            C                                              (        17        )            
It can be shown that the inequalities (17) make it necessary and sufficient for the integer components of b to satisfy:
                                                        ⌈                                                -                                                            C                                              q                                                                              2                            ⁢                            M                                                    ,                                                      2                            ⁢                            M                                                                                                                                              +                                  ρ                                      2                    ⁢                    M                                                              ⌉                        ≤                          b                              2                ⁢                M                                      ≤                                          ⌊                                                                            C                                              q                                                                              2                            ⁢                            M                                                    ,                                                      2                            ⁢                            M                                                                                                                                +                                      ρ                                          2                      ⁢                      M                                                                      ⌋                            ⁢                              ⌈                                                      -                                                                                            C                          -                                                                                    q                                                                                                2                                  ⁢                                  M                                                                ,                                                                  2                                  ⁢                                  M                                                                                                                      ⁢                                                          ξ                                                              2                                ⁢                                M                                                            2                                                                                                                                q                                                                                                                    2                                ⁢                                M                                                            -                              1                                                        ,                                                                                          2                                ⁢                                M                                                            -                              1                                                                                                                                                            +                                      ρ                                                                  2                        ⁢                        M                                            -                      1                                                        +                                                            q                                                                        2                          ⁢                          M                                                ,                                                                              2                            ⁢                            M                                                    -                          1                                                                                      ⁢                                          ξ                                              2                        ⁢                        M                                                                                            ⌉                                      ≤                          b                                                2                  ⁢                  M                                -                1                                      ≤                                          ⌊                                                      -                                                                                            C                          -                                                                                    q                                                                                                2                                  ⁢                                  M                                                                ,                                                                  2                                  ⁢                                  M                                                                                                                      ⁢                                                          ξ                                                              2                                ⁢                                M                                                            2                                                                                                                                q                                                                                                                    2                                ⁢                                M                                                            -                              1                                                        ,                                                                                          2                                ⁢                                M                                                            -                              1                                                                                                                                                            +                                      ρ                                                                  2                        ⁢                        M                                            -                      1                                                        +                                                            q                                                                        2                          ⁢                          M                                                ,                                                                              2                            ⁢                            M                                                    -                          1                                                                                      ⁢                                          ξ                                              2                        ⁢                        M                                                                                            ⌋                            ⁢                              ⌈                                                      -                                                                                            1                                                      q                            ii                                                                          ⁢                                                  (                                                      C                            -                                                                                          ∑                                                                  l                                  =                                                                      i                                    +                                    1                                                                                                                                    2                                  ⁢                                  M                                                                                            ⁢                                                                                                                                    q                                    ll                                                                    ⁡                                                                      (                                                                                                                  ξ                                        l                                                                            +                                                                                                                        ∑                                                                                      j                                            =                                                                                          l                                              +                                              1                                                                                                                                                                            2                                            ⁢                                            M                                                                                                                          ⁢                                                                                                                              q                                            jl                                                                                    ⁢                                                                                      ξ                                            j                                                                                                                                                                                                )                                                                                                  2                                                                                                              )                                                                                                      +                                      ρ                    i                                    +                                                            ∑                                              j                        =                                                  i                          +                          1                                                                                            2                        ⁢                        M                                                              ⁢                                                                  q                        ji                                            ⁢                                              ξ                        j                                                                                            ⌉                                      ≤                          b              i                                ⁢                                          ⁢                                          ⁢                                          ⁢                                    b              i                        ≤                          ⌊                                                                                          1                                              q                        ii                                                              ⁢                                          (                                              C                        -                                                                              ∑                                                          l                              =                                                              i                                +                                1                                                                                                                    2                              ⁢                              M                                                                                ⁢                                                                                                                    q                                ll                                                            ⁡                                                              (                                                                                                      ξ                                    l                                                                    +                                                                                                            ∑                                                                              j                                        =                                                                                  l                                          +                                          1                                                                                                                                                            2                                        ⁢                                        M                                                                                                              ⁢                                                                                                                  q                                        jl                                                                            ⁢                                                                              ξ                                        j                                                                                                                                                                            )                                                                                      2                                                                                              )                                                                      +                                  ρ                  i                                +                                                      ∑                                          j                      =                                              i                        +                        1                                                                                    2                      ⁢                      M                                                        ⁢                                                            q                      ji                                        ⁢                                          ξ                      j                                                                                  ⌋                                      ⁢                                                      (        18        )            where ┌x┐ is the smallest integer greater than the real number x and └x┘ is the largest integer smaller than the real number x .
In practice 2M internal counters are used, namely one counter per dimension, each counter counting between a lower and an upper bound as indicated in (18), given that each counter is associated with a particular pair of bounds. In practice these bounds can be updated recursively. We put:
                              S          i                =                                            S              i                        ⁡                          (                                                ξ                                      i                    +                    1                                                  ,                …                ⁢                                                                  ,                                  ξ                                      2                    ⁢                    M                                                              )                                =                                    ρ              i                        +                                          ∑                                  j                  =                                      i                    +                    1                                                                    2                  ⁢                  M                                            ⁢                                                q                  ji                                ⁢                                  ξ                  j                                                                                        (        19        )                                          T                      i            -            1                          =                                            T                              i                -                1                                      ⁡                          (                                                ξ                  i                                ,                …                ⁢                                                                  ,                                  ξ                                      2                    ⁢                    M                                                              )                                =                                    C              -                                                ∑                                      l                    =                    i                                                        2                    ⁢                    M                                                  ⁢                                                                            q                      ll                                        ⁡                                          (                                                                        ξ                          l                                                +                                                                              ∑                                                          j                              =                                                              l                                +                                1                                                                                                                    2                              ⁢                              M                                                                                ⁢                                                                                    q                              jl                                                        ⁢                                                          ξ                              j                                                                                                                          )                                                        2                                                      =                                          T                i                            -                                                                    q                    ii                                    ⁡                                      (                                                                  ξ                        i                                            +                                              S                        i                                            -                                              ρ                        i                                                              )                                                  2                                                                        (        20        )            Ti−1=Ti−qii(Si−bi)2  (21)
with T2M=C.
Using equations (19) to (21), the range of variation of each component bi is determined recursively, commencing with the component b2M:Li−≦bi<Li+  (22)with
                              L          i          -                =                                            ⌈                                                -                                                                                    T                        i                                                                    q                        il                                                                                            +                                  S                  i                                            ⌉                        ⁢                                                  ⁢            and            ⁢                                                  ⁢                          L              i              +                                =                      ⌊                                                                                T                    i                                                        q                    ii                                                              +                              S                i                                      ⌋                                              (        23        )            
For each candidate vector b (in the representation of FIG. 2B), it is checked whether b∈Π. Once the closest vector b is found, the estimates of the real and imaginary parts the transmitted symbols are simply obtained from (6):{circumflex over (x)}′=2b−μ  (24)
As described above, in relation with expressions (18) to (23), the sphere decoding process necessitates to go through every lattice point included in the ellipsoid defined by (17) and to compute for each of them the norm ∥w∥. This scanning algorithm is very time consuming, in particular for a large number of transmit antennas M.
In order to speed up the sphere decoding process it has already been proposed to update the radius √{square root over (C)} with the lowest computed norm ∥w∥, thereby scaling down the ellipsoid each time a lower norm is found. Each resealing implies however the computation of new bounds Li− and Li+ and is followed by a new search between the updated bounds. It is therefore desirable to reduce the complexity of a sphere decoding method in a receiver of a MIMO telecommunication system for obtaining an estimation of the transmitted symbols.
FIG. 3A illustrates schematically an example of a space-time encoder 110, as disclosed in the article of B. Hassibi et al. in the article entitled “High-rate codes that are linear in space and time” available on the website mars.bell-labs.com.
The input data stream Din input to the space-time encoder is multiplexed by a multiplexer 111 in a plurality Q of bit substreams which are respectively mapped by the symbol mappers 1121, . . . , 112Q into Q symbol substreams. A space-time dispersion symbol encoder 113, the function of which is hereinafter described, transforms a block of Q uncoded symbols, denoted x1u, . . . , xQu into a frame of L consecutive vectors (x1(1), . . . , xM(1) ), (x1(2), . . . , xM(2)), . . . , (x1(L), . . . , xM(L)), each vector being transmitted by the M transmit antennas 1301, 1302, . . . , 130M. In general, the symbols x1(p), . . . , xM(p) transmitted at a given time p=1, . . . , L are not independent. More precisely, each uncoded symbol xqu is coded in 113 by two L×M spatio-temporal dispersion matrices Aq and Bq as a combination xqu.Aq+xqu*Bq (where .* denotes the conjugate) and the block of symbols x1u, . . . , xQu is coded as the sum:
                    X        =                              ∑                          q              =              1                        Q                    ⁢                                          ⁢                      (                                                            x                  q                  u                                .                                  A                  q                                            +                                                x                  q                                      u                    *                                                  ⁢                                  B                  q                                                      )                                              (        25        )            where the rows of the matrix X are the vectors of the transmitted symbols at the different times p=1, . . . , L.
FIG. 3B illustrates schematically the structure of the detector 160 when the space-time encoder of FIG. 3A is used. A symbol decoder 161 inputs the L consecutive vectors (y1(1), . . . , yN(1)), (y1(2), . . . , yN(2)), . . . , (y1(L), . . . , yN(L)) each vector being relative to a reception time p=1, . . . , L and constituted of the signals received by the N receive antennas 1401, 1402, . . , 140N at that time.
The L consecutive vectors can be represented as rows of a L×N matrix Y. We have:Y=XH+Ξ  (26)where Ξ is a L×N matrix the rows of which represent the complex noise samples affecting the received symbols for the L consecutive timeslots.
In practice, for a given number N of transmit antennas, the block size Q and the frame length L are chosen such that Q≦L.N in order to avoid that the linear system (26) be undetermined in x1u, . . . , xQu. It should also be noted that if Q<L.M, some redundancy is introduced at the symbol coding level by the symbol encoder 113.
It can be shown that the expression (26) can equivalently be reformulated as follows:y′Lxu′H′L+η′L  (27)where y′L is the 1×2LN vector obtained by concatenating the L consecutive vectors y′(p)=(y1R(p), y1I(p), . . . , yNR(p), yNI(p)), p=1, . . . , L; η′L is the 1×2LN vector obtained by concatenating the L consecutive vectors η′(p)=(η1K(p), η1I(p), . . . , ηNR(p), ηNI(p)) p=1, . . . , L; xu′ is the 1×2Q vector
  (            x      1      uR        ,                  x        1        ul            ⁢                          ⁢      …        ⁢                  ,          x      Q      uR        ,          x      Q      ul        ,    )where
      x    q    u    =            x      q      uR        +          j      ⁢                          ⁢              x        q        ul            
and H′L is a 2Q×2LN matrix of real coefficients obtained from the (real and imaginary parts of the) matrix H of the fading coefficients and the (real and imaginary parts of the) dispersion matrices Aq and Bq, q=1, . . . , Q. If Q≦L.N and the dispersion matrices are properly chosen, the matrix is not degenerate.
The expression (27) is similar to (3), though it involves higher vector and matrix dimensions: the number (N) of receive antennas is multiplied by the number (L) of timeslots and the number (M) of transmit antennas is replaced by the block size (Q) of the uncoded symbols. The product constellation Π is generated by the constellation modulations used for mapping the different symbols of the block.
The symbol decoder 161 may therefore carry out a sphere decoding method as set out above, the lattice being here generated by the generator matrix H′L instead of the matrix H′, and therefore has a dimension κ=2Q in a space of dimension K=2LN. The sphere decoding algorithm provides a vector {circumflex over (x)}u′ (from an expression similar to (24)) which gives the estimates
            x      ^        1    uR    ,                    x        ^            1      ul        ⁢                  ⁢    …    ⁢          ,            x      ^        Q    uR    ,            x      ^        Q    ul    ,
i.e. the complex estimates {circumflex over (x)}qu of the uncoded symbols xqu. The Q estimates are demapped by the demappers 1621, . . . , 162Q and the binary substreams thus generated are demultiplexed by a demultiplexer 163 to produce a stream of received data Dout.
It should be noted that when a space-time dispersion symbol encoder is used, the sphere decoding method enables the detection of the symbols xqu input to the encoder and not of the symbols xm as transmitted.
If the block size Q is large, scanning the candidate lattice points within the ellipsoid, along each of the 2Q dimensions of the lattice (M is to be replaced by Q in the expressions (18)) may be very time consuming.